of Sampling continuous-time signals
Chap: 4.1-4.8; Pages: 140-201
Jont Allen
ECE-310
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Periodic symmetry
Every function may be made -periodic with an overlap and add OLA operation
and integer, the period
Functions periodic in one domain (e.g., time) are sampled in the other domain (e.g., frequency) Convergence of this expression is an issue
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Review of nomenclature
FT type DTFT 48
time
limits
freq.
limits
in ROC
z Transform 95
Fourier Transform 143
C/D Transform 143
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Periodic sampling 140
Starting from a continuous-time signal , a sampler determines a discrete-time signal
C/D
is sampled
periodic in frequency
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Periodic sampling
Every periodic function may be expanded in harmonics, at frequencies
From Fourier series formula:
Periodic impulses: page 143, Eq. 4.5
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Basic symmetry
Periodic impulses 143, Eq. 4.5 and Munson notes 25.2
This is the Poisson Summation Formula PSF PSF is a very important result Based on Fourier series expansion of impulse-train
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Applications of PSF
Let
.
Modulation formula:
on right
Multiply by on left, convolve Overlap-add formula:
Convolve on left, multiply
on right
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Frequency domain nomenclature
Details in working with and :
A frequency of Hz has a radian frequency of , which corresponds to a normalized . frequency of Delta scaling for any an example. Example 4.1 148
. Try
as
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Pulse train modulation
General case of time modulation
Image
Continuous time Fourier Transform pair
Discrete time DTFT pair
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Effect of increased sampling rate
When is halved (
,
doubled,):
the images move out:
Image
Image
Discrete time
Discrete time
1X SAMPLING
2X SAMPLING
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Effect of decreased sampling rate
When is doubled (
,
halved):
the images move in. Overlap in the spectrum
is called aliasing
Image
Image
Discrete time
Discrete time
1X SAMPLING
0.5X SAMPLING
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Harry Nyquist
Born in Sweden; Three famous theorems named after him: The Nyquist 1. Sampling Thm., (Nyquist 1928) 2. Thermal noise Thm., (Nyquist 1932) and 3. Feedback stability Thm. (Nyquist 1934)
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Windowing the images
What happens when images are removed by windowing in frequency?
Image
Continuous time Fourier Transform pair
Discrete time DTFT pair
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Nyquist sampling theorem 1928
Any signal may be uniquely represented by its samples if it is sampled at , dened as more than twice its highest frequency 146
Note the somewhat confusing denitions in the book 146 regarding the terms Nyquist frequency , versus the Nyquist rate .
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Some issues to think about
The proof of the Sampling Theorem is based on convolution with , namely the formula: 150
reconstructed
This low-pass reconstruction lter
is also called the interpolation lter, as it interpolates the signal between samples.
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Some issues to think about
In practice convolution by a perfect lter
reconstructed
is noncausal, and therefore cannot be implemented. A casual low-pass lter is used in practice. What are the practical implications of this? How will differ from the starting at the input to the ideal C/D followed by D/C conversion process? Namely what is the RMS error going to look like? In practice this works because the ear cannot hear the phase distortion
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Poisson Summation Formula
Case of impulse:
General case of OLA comes from convolution on left by :
is , sampled at
is a continuous time function
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Nyquists famous problem
Find the Johnson thermal noise Transmission line terminated in resistors
Length
Stored energy
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Nyquists 2 famous problem
At
, remove the resistors
Length
Stored energy
Nyquists Johnson-noise formula follows:
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DT processing of CT signals 4.4
Basic model of C/D D/C processing C/D/C 153-154
Discretetime system
C/D
D/C
Ideal reconstruction (antialias) lter in D/C
This basic structure describes almost every telephone, for more than 30 years
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Two basic type of C/D/C systems
There are two basic categories of C/D/C systems: Real-time processing: Any application where 1 sample in gives 1 sample out is a real-time method Non-real-time processing: non-real-time I applications are those where input time and output time are different, or non-real-time II where the computation time takes so much time that the processing cannot keep up
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Real-time (RT) processing examples
Today there are a great many examples of real-time systems: cell phones, CD players, hearing aids, video conferencing over a wide-band channel (i.e., the Internet) Sometimes we nd these systems to fall out of real time (i.e., cell phones and video)
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Non-real-time (NRT) processing examples
Cases of non-real-time applications where the input and output rates differ: Examples: pagers, fax machines, . . . These make you wait
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Non-real-time processing examples II
Cases of non-real-time applications where the computation time is greater than real-time video-conferencing over phone lines, cell-3G (3 generation cell), some classes of speech-De-noising and music encoding such as MPEG audio and video
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OLA processing
For many C/D/C processing schemes RT and NRT, OLA frequency domain processing is used This method is based on the OLA formula
Let
be a low-pass lter. small such that
for
.
Under these conditions
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OLA formula
From the last slide:
Typically
, where is the length of window , Expand signal into OLA smooth blocks
Dene the windowed signal as
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From frequency to time by OLA
Expand signal into smoothed OLA blocks
As before:
where
Thus
: Fourier Transform
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Aliasing 4.1-4.3 147-149
Example of aliasing of a 110 Hz tone:
Aliasing: every 11 sample of y[n] 1 y[n]
th
0
1 0
2 1
0.2 F
100 50
0.4 0.6 Time [s]
max
0
0.8 F
max
1
110 [Hz]
10
0 150
50
100
150
Frequency [Hz]
Sample period [s] [Hz]: [Hz] stems:
Hz;
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Aliasing 4.1-4.3 147-149
Example of decimation-aliasing of a tone:
Aliasing: every 11 sample of y[n] 1 y[n]
th
0
1 0
2 1
0.2 F
100 50
0.4 0.6 Time [s]
max
0
0.8 F
max
1
110 [Hz]
10
0 150
50
100
150
Frequency [Hz]
Sample period [s] Hz; [Hz] For the blue curve .
Thus
,
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Down-sampling 158
Suppose we cut the bandwidth by 2 in the frequency domain with an ideal low-pass lter We may then reduce
at the output by 2x, otherwise
without aliasing, thus
Alternate samples are dropped in this processing, called down-sampling
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Ideal differentiator 158
Suppose we wish to differentiate a continuous input signal
This causal frequency response corresponds to
Thus
eff
It may be shown that note the noncausal nature of
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Impulse-invariance 160
Relate and
to
Discretetime system
C/D
D/C
,
eff
Continuoustime LTI system
With impulse invariance the mapping from discrete to the continuous domain is dene by
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Example of impulse-invariance 162
A common continuous-time impulse response, and corresponding Laplace Transform:
From impulse-invariance note error in book 4.8 162
This common example must alias since is not bandlimited
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Upsampling by linear interpolation I
When upsampling, we need to interpolate the new samples (Matlab help upsample, interp)
1 0
1 1
0
5
10
15
20
25
30
0
1 1
0
10
20
30
40
50
60
y[n+1] =(y[n]+y[n+2])/2
0
1 0.05 Error
0
10
20
30
40
50
60
0
0.05
0
10
20
30
40
50
60
This may be done by linear interpolation, but at a cost.
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Upsampling by linear interpolation II
Frequency response of a linear interpolator
1 0
1 1
0
5
10
15
20
25
30
0
1 1
0
10
20
30
40
50
60
y[n+1] =(y[n]+y[n+2])/2
0
1 0.05
0
10
20
30
40
50
60
Error
0
0.05
0
10
20
30
40
50
60
Even stems from linear interpolation Red curve is the exact; ): The error is of the form (note
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Upsampling by linear interpolation III
Distortion is audible
F =1e4 sin(2 1000 n T)
10 10 10 10 10 10
20
s
0
f
10
3
0
Fs=10
4
20 20
10
4
lin interpolated
0
f
10
3
0
F =2 104
s
F f
s
0
10
4
20
error=|S(f)S (f)|
i
|error| = 5%
10
0 3
2 Fs=10
10
4
4
10
FREQUENCY (Hz)
The error due to interpolation is 5% at There is an unwanted tone at
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DT processing of analog signals 4.8 185
DT processing with A/D and D/A
C/D
Discretetime system
D/C
Sample and hold
Anti aliasing filter
A/D converter
Discretetime system
D/A converter
Compensated reconstruction
filter
T
T
T
Basic signal denitions
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Traditional C/D conversion
Traditional converter requires a high order lter
[dB]
Spectral noise floor
Traditional filter
[dB]
Shaped noise floor
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Some issues
Analog prelter is very expensive, large, requires laser trimmed resistors Phase response distorts the waveform (not necessarily a problem) Needs to be a very large order (i.e., >100 dB/oct) The oversampled sigma-delta - converter solved all these proble...